Worked solutions for example 4 Workshop 2 - Basic Algebra OES Contextual problems

1. 4a Using the other units given in the question, deduce the
units of wind stress. What else is this unit known as?
What is it a measure of?

2. 4a Note: Brackets around 휏 represent its units.
• Replace the values in the first equation for their units:

3. 4a Note: Brackets around 휏 represent its units.
• Replace the values in the first equation for their units:
(τ) = kgm−3. ms−1. ms−1

4. 4a Note: Brackets around 휏 represent its units.
• Replace the values in the first equation for their units:
(τ) = kgm−3. ms−1. ms−1
• Group the same units together and simplify:

5. 4a Note: Brackets around 휏 represent its units.
• Replace the values in the first equation for their units:
(τ) = kgm−3. ms−1. ms−1
• Group the same units together and simplify:
(τ) = kg (m−3. m1. m1)(s−1. s−1)

6. 4a Note: Brackets around 휏 represent its units.
• Replace the values in the first equation for their units:
(τ) = kgm−3. ms−1. ms−1
• Group the same units together and simplify:
(τ) = kg (m−3. m1. m1)(s−1. s−1)
(τ) = 퐤퐠퐦−ퟏ퐬−ퟐ

7. 4a Note: Brackets around 휏 represent its units.
• Replace the values in the first equation for their units:
(τ) = kgm−3. ms−1. ms−1
• Group the same units together and simplify:
(τ) = kg (m−3. m1. m1)(s−1. s−1)
(τ) = 퐤퐠퐦−ퟏ퐬−ퟐ
This unit is also known as a Pascal, Pa. It is a unit of pressure.

8. 4b What wind speed would produce a drag coefficient of 1.05
x 10-3?

9. 4b • Substitute the values you know into the second equation:

10. 4b • Substitute the values you know into the second equation:
1.05 x 10−3 = 0.53 + 0.064U x 10−3

11. 4b • Substitute the values you know into the second equation:
1.05 x 10−3 = 0.53 + 0.064U x 10−3
• Cancel out 10-3:

12. 4b • Substitute the values you know into the second equation:
1.05 x 10−3 = 0.53 + 0.064U x 10−3
• Cancel out 10-3:
1.05 = 0.53 + 0.064U

13. 4b • Substitute the values you know into the second equation:
1.05 x 10−3 = 0.53 + 0.064U x 10−3
• Cancel out 10-3:
1.05 = 0.53 + 0.064U
• Rearrange for U:

14. 4b • Substitute the values you know into the second equation:
1.05 x 10−3 = 0.53 + 0.064U x 10−3
• Cancel out 10-3:
1.05 = 0.53 + 0.064U
• Rearrange for U:
1.05 − 0.53 = 0.064U

15. 4b • Substitute the values you know into the second equation:
1.05 x 10−3 = 0.53 + 0.064U x 10−3
• Cancel out 10-3:
1.05 = 0.53 + 0.064U
• Rearrange for U:
1.05 − 0.53 = 0.064U
U =
1.05 − 0.53
0.064
= ퟖ. ퟏퟐퟓ퐦퐬−ퟏ

16. 4c Using only the first equation for wind stress, calculate the
wind speed needed to produce a wind stress of 4.9 x 10-2
Pa. Assume the density of air is 1.225 kgm3 and the drag
coefficient is 9.46 x 10-4.

17. 4c • Rearrange the first equation for U:

18. 4c • Rearrange the first equation for U:
ρACDU2 = τ

19. 4c • Rearrange the first equation for U:
ρACDU2 = τ
U2 =
τ
ρACD

20. 4c • Rearrange the first equation for U:
ρACDU2 = τ
U2 =
τ
ρACD
U =
τ
ρACD

21. 4c • Rearrange the first equation for U:
ρACDU2 = τ
U2 =
τ
ρACD
U =
τ
ρACD
• Substitute in the values you know:

22. 4c • Rearrange the first equation for U:
ρACDU2 = τ
U2 =
τ
ρACD
U =
τ
ρACD
• Substitute in the values you know:
4.9 x 10−2
1.225 x 9.46 x 10−4 = ퟔ. ퟓퟎ퐦퐬−ퟏ